Преобразование выражений, содержащих степени. Задача 45math100admin44242025-03-26T20:47:30+03:00
Задача 45. Вычислите \(\dfrac{{5 \cdot {2^{13}} \cdot {4^{11}}-{{16}^9}}}{{{{\left( {{2^{11}}} \right)}^3}}} + \dfrac{{{2^{21}} \cdot {{27}^3} + 15 \cdot {4^{10}} \cdot {9^4}}}{{{6^9} \cdot {2^{10}} + {{12}^{10}}}}\)
Решение
\(\dfrac{{5 \cdot {2^{13}} \cdot {4^{11}}-{{16}^9}}}{{{{\left( {{2^{11}}} \right)}^3}}} + \dfrac{{{2^{21}} \cdot {{27}^3} + 15 \cdot {4^{10}} \cdot {9^4}}}{{{6^9} \cdot {2^{10}} + {{12}^{10}}}} = \dfrac{{5 \cdot {2^{13}} \cdot {2^{22}}-{2^{36}}}}{{{2^{33}}}} + \dfrac{{{2^{21}} \cdot {3^9} + 15 \cdot {2^{20}} \cdot {3^8}}}{{{3^9} \cdot {2^9} \cdot {2^{10}} + {3^{10}} \cdot {4^{10}}}} = \)
\( = \dfrac{{5 \cdot {2^{35}}-2 \cdot {2^{35}}}}{{{2^{33}}}} + \dfrac{{{2^{20}} \cdot {3^8} \cdot \left( {2 \cdot 3 + 15} \right)}}{{{3^9} \cdot {2^{19}} + 3 \cdot {3^9} \cdot 2 \cdot {2^{19}}}} = \dfrac{{{2^{35}} \cdot \left( {5-2} \right)}}{{{2^{33}}}} + \dfrac{{{2^{20}} \cdot {3^8} \cdot 21}}{{{3^9} \cdot {2^{19}} \cdot \left( {1 + 6} \right)}} = \)
\( = {2^2} \cdot 3 + \dfrac{{2 \cdot 21}}{{3 \cdot 7}} = 12 + 2 = 14.\)
Ответ: 14.